Random intercepts model using hiv-intervention data

Topics to be covered

What you will learn
- Mathematical formulation of random intercepts model
- Description of HIV-intervention data
- Random intercepts model using hiv-intervention data
- Mathematical formulation of random slopes model
- Assumptions and complications

Longitudinal data

Measurements taken at different times
- Emphasis in changes over time

In the previous module, I talked about hierarchical models and mentioned a particular case, longitudinal data, that I want to talk more about in this presentation.

Longitudinal data is similar to repeated measures data. With both, you measure the same subject repeatedly. With longitudinal data, often the emphasis is in changes that occur over time. Repeated measurements, in contrast, emphasize different treatments with the hope that the time gaps between the measurements are small enough that you don’t see changes over time.

The differences between longitudinal data, repeated measures data, or hierarchical data are subtle. Perhaps these are distinctions without a difference. I decided to separate out longitudinal data for a different module perhaps more out of the desire to split a complex topic into smaller bite-sized pieces.

Random intercepts model, 1 of 3

Simplest pattern for longitudinal data
\(Y_{ij},\ i=1,...,n;\ j=1,...,k\)
- n subjects, k time points
\(t_j\), time of jth measurement
- First time is often zero

Random intercepts model, 2 of 3

\(Y_{ij}=\beta_0+u_{0i}+\beta_1 t_j + \epsilon_{ij}\)
- \(\beta_0\) and \(\beta_1\) are unknown constants
- \(u_{0i}\) and \(\epsilon_{ij}\) are normally distributed
  - \(SD(u_{0i})=\sigma_{intercept}\)
  - \(SD(\epsilon_{ij})=\sigma_{error}\)

Random intercepts model, 3 of 3

\(SD(Y_{ij})=\sqrt{\sigma^2_{intercept}\ +\ \sigma^2_{error}}\)
\(Corr(Y_{ij}, Y_{im})=\frac{\sigma^2_{intercept}}{\sigma^2_{intercept}\ +\ \sigma^2_{error}}\)

Random intercepts illustrated, 1 of 2

Random intercepts illustrated, 2 of 2

Break #1

What you have learned
- Mathematical formulation of random intercepts model
What’s coming next
- Description of HIV-intervention data

Description of hiv-intervention data, 1 of 2

data_dictionary: hiv-intervention.txt

source: OzDASL website

description: |
  This is a longitudinal study of an intervention in 14-18 adolescents  intended to increase the frequency of condom protected sex. Subjects  were allocated randomly to treatment or control groups. All were evaluated prior to the intervention, immediately  after the intervention, 6 months and  12 months after the intervention.The outcome variable is the logarithm-transformed frequency of condom-protected sex ( log(Y+1) )."

Description of hiv-intervention data, 2 of 2

BST:
  label: treatment group
  values:
    '1': BST intervention
    '0': control
Pre:
  label: Log-frequency of protected sex before the intervention
Post:
  label: Log-frequency of protected sex after the intervention
FU6:
  label: Log-frequency of protected sex reported at the 6 months follow-up
FU12:
  label: Log-frequency of protected sex reported at the 12 months follow-up

Wide format

Partial listing of original data with measurement times in separate columns

Boxplots

Colors and patterns

Dialog box showing color and pattern choices

Tall format

Partial listing of restructured data with all time measurements in a single column

Alternate clustering of boxplots

Boxplots with treatment groups side by side

Live demo, restructuring and boxplots

After menu Graph | Boxplot

Boxplot dialog box with icon Clustered and option Summaries of separate variables

After button Define

Summaries of Separate Variables dialog box with field Boxes Represent: Pre, Post, FU6, FU12 and field Category Axis: BST

After menu Data | Restructure

Restructure Data Wizard dialog box with option Restructure selected variables into cases

After button Next

Step 2 dialog box with default option One

After button Next

Step 3 dialog box with field Name id, field Target Variable: log_frequency and field Target Variable: Post, FU6, FU12, and field Fixed Variable(s): BST, Pre

After button Next

Step 4 dialog box with default option One

After button Next

Step 5 dialog box with option Variable names

After menu Transform | Recode into Different Variables

Recode into Different Variables dialog box with field String Variable: Index1 and field Output Variable Name: time

After button Old and New Values

Old and New Values dialog box with Old --> New field: MISSING -> SYSMIS, 'Post' -> 0, 'FU6 -> 6, 'FU12' -> 12, ELSE -. 99

After menu Graphs | Boxplot

Boxplot dialog box with icon Clustered and option Summaries for groups of cases

After button Define

Summaries for Groups of Cases dialog box with field Variable log_frequency, field Category Axis: time, and field Define Clusters by: BST

Break #2

What you have learned
- Description of HIV-intervention data
What’s coming next
- Random intercepts model using hiv-intervention data

Random intercepts analysis, 1 of 6

Random intercepts analysis, 2 of 6

Random intercepts analysis, 3 of 6

Random intercepts analysis, 4 of 6

Random intercepts analysis, 5 of 6

Random intercepts analysis, 6 of 6

itle: “Restructure demo” ormat: revealjs: slide-number: true embed-resources: true ditor: source

Live demo, Random Intercepts Model

After menu Analyze | Mixed Models | Linear

Specify Subjects and Repeated dialog box with field Subjects: id

After button Continue

Linear Mixed Models dialog box with field Dependent Variable: log_frequency, field Factor(s): BST, and field Covariate(s): time

After button Fixed

Fixed Effects dialog box with field Model: BST, time and option Include intercept

After button Random

Random Effects dialog box with field Combinations: id and option Display parameter predictions

After button Statistics

Statistics dialog box with option Parameter estimates for fixed effects

After button Save

Save dialog box with second option Predicted Values and option Residuals

After button Export

Export dialog box with option Export BLUPS

Break #3

What you have learned
- Random intercepts model using hiv-intervention data
What’s coming next
- Mathematical formulation of random slopes model

title: “Mathematical formulation of the random slopes model” format: revealjs: slide-number: true embed-resources: true editor: source

Random slopes model, 1 of 2

Same notation for the time and outcome variables
\(Y_{ij},\ i=1,...,n;\ j=1,...,k\)
- n subjects, k time points
\(t_j\), time of jth measurement

Random slopes model, 2 of 2

\(Y_{ij}=\beta_0+u_{0i}+\beta_1 t_j+u_{0i} t_j+\epsilon_{ij}\)
- \(\beta_0\) and \(\beta_1\) are unknown constants
- \(u_{0i}\), \(u_{0i}\), and \(\epsilon_{ij}\) are normally distributed
  - \(SD(u_{0i})=\sigma_{intercept}\)
  - \(SD(u_{1i})=\sigma_{slope}\)
  - \(SD(\epsilon_{ij})=\sigma_{error}\)

Random slopes illustrated, 1 of 2

Random slopes illustrated, 2 of 2

Break #4

What you have learned
- Mathematical formulation of random slopes model
What’s coming next
- Assumptions and complications

title: “Assumptions and complications” format: revealjs: slide-number: true embed-resources: true editor: source

Assumptions

Independence
- Only between subjects
Normality
- Residuals
- Random intercepts and/or slopes
Linearity

Normality check, 1 of 2

Normal probability plot shows close to a straight line indicating normality of the random intercepts

Normality check, 2 of 2

Normal probability plot shows a slight curvature indicating possible non-normality of the residuals

Linearity check

Complications

Not a problem
- Missing values
- Better than Last Observation Carried Forward
Problems (more tedious than difficult)
- Interactions
- Nonlinear trends
- Covariates
  - Between patients
  - Within patients

Missing values are normally a big headache, but less so for the random intercepts or slopes models. If a patient missed a visit at a particular time point, you can easily extrapolate from the visits before and after. You are assuming linearity, after all.

In some clinical trials, if a patient came at the intermediate evaluation but did not show up at the final evaluation, the research team would replace the missing final outcome with the intermediate outcome. This technique, called Last Observation Carried Forward (LOCF) was not very popular when it was introduced and has been pretty much discredited. The random intercepts or slopes model would extrapolate the trend from the intermediate value, which works much better.

Interactions are always difficult, and when the interactions involve time, it can get a bit messy. Nonlinear trends over time are also a bit of a problem. In both cases, though, the work is more tedious than difficult. Interactions and nonlinearity mean that your interpretation of the results will require a bit more thought and you can’t come up with as simple a story to tell.

A covariate is a variable which is not of direct interest in the research, but one that you must take account of in order to produce a credible analysis. In just about any cancer study, you should track whether the patient is a smoker. It’s not something you’re interested in testing. The role of smoking in lung cancer and most other types of cancer was established many decades ago. You still have to account for smoking though because it can explain so much of the variation in your outcome. Failure to account for smoking would greatly reduce your power and precision.

There are two types of covariates. The ones that are fixed and do not change over time are called time constant covariates or between subject covariates. Patient demographics are time constant. Measurements done at baseline to assess how ill the patient was at the start of the study are time constant.

Covariates that change over time are called time varying covariates or within subject covariates. The extent to which a patient complies with taking his/her medication is a time varying covariate. Seasonal changes in temperature, humidity, or pollen counts are time varying covariates.

There is one important distinction between time constant and time varying covariates. The latter are much better at removing variation from your outcome, and can greatly improve your power and precision.

Live demo, normality checks

Summary

What you have learned
- Mathematical formulation of random intercepts model
- Description of HIV-intervention data
- Random intercepts model using hiv-intervention data
- Mathematical formulation of random slopes model
- Assumptions and complications

Additional topics??

Learning objectives for mixed models
- Set up a data set for a Repeated Measures ANOVA
- Explain when a n-way RM ANOVA would be used to analyze data
- Explain the data requirements for a n-way RM ANOVA
- Write a research question for a n-way RM ANOVA
- Write the Null and Research Hypotheses for a n-way RM ANOVA
- Run a complete n-way RM ANOVA
- List the assumptions for a n-way RM ANOVA
- Explain the tests for the assumptions for a n-way RM ANOVA
- Run the assumptions tests for a n-way RM ANOVA
- Interpret the assumptions tests for a n-way RM ANOVA
- Explain the issue that arises if the assumption of sphericity is violated
- Explain how the Epsilon correction works
- Explain the condition of a significant interaction effect
- Explain the process to run simple main effects for a Repeated Measures n-way ANOVA
- Run simple main effects for a Repeated Measures n-way ANOVA
- Interpret simple main effects for a Repeated Measures n-way ANOVA
- Explain the special considerations for simple main effects for a Repeated Measures n-way ANOVA with a factor with >2 levels
- Explain the process to run main effects for a Repeated Measures n-way ANOVA
- Run main effects for a Repeated Measures n-way ANOVA
- Interpret main effects for a Repeated Measures n-way ANOVA
- Summarize the output of a complete n-way RM ANOVA
- Define the meaning of mixed model
- Define Between Subjects in relation to the mixed model
- Define Within Subjects in relation to the mixed model
- Be familiar with the various names for a mixed model ANOVA
- List the purpose of running a mixed model ANOVA
- Identify Between and Within factors for a mixed model ANOVA
- List the type of data that can be used to define the levels in a mixed model ANOVA
- List the assumptions for a mixed model ANOVA
- Successfully run the statistical analysis tests for the assumptions of a mixed model ANOVA
- Interpret the statistical analysis tests for the assumptions of a mixed model ANOVA
- Explain how normality of the data is assessed
- Explain the rules for violation of the assumption of homogeneity of variances
- Define Homogeneity of Covariances
- Identify Box’s Test
- Develop a research question for a mixed model ANOVA
- Write the Null and Alternative for a mixed model ANOVA
- Set up data for a mixed model ANOVA in the SPSS data editor
- Run a two-way mixed model ANOVA analysis using SPSS
- Interpret the output of a two-way mixed model ANOVA analysis
- Understand the process of determining the presence or not of an interaction effect
- Define Simple Main Effects
- Determine when Simple Main Effects would be run
- Define Main Effects
- Determine when Main Effects would be run
- Run the Simple Main Effects following a decision about the interaction effect
- Interpret the Simple Main Effects
- Run the Main Effects following a decision about the interaction effect
- Interpret the Main Effects
- Interpret Pair-Wise comparisons
- Create a complete write up of a mixed model ANOVA analysis

title: “simon-5502-12-slides”